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Inverse Functions Part 1

Inverse Functions Part 1. Goal: Find inverses of linear functions. Inverse Functions. A function and its inverse can be described as the "DO" and the "UNDO" functions.  A function takes a starting value, performs some operation on this value, and creates an output answer. 

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Inverse Functions Part 1

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  1. Inverse FunctionsPart 1 Goal: Find inverses of linear functions.

  2. Inverse Functions • A function and its inverse can be described as the "DO" and the "UNDO" functions.  • A function takes a starting value, performs some operation on this value, and creates an output answer.  • The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

  3. Definition of an Inverse •  The inverseof a functionis the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. Notation:   If  f (x) is a given function, then f-1(x) denotes the inverse of  f.

  4. Finding the Inverse of a Function • Basically, the process of finding an inverse is simply the swapping of the x and y coordinates.  • This newly formed inverse will be a relation, but may not necessarily be a function. 

  5. Horizontal Line Test The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more that once.

  6. Three Methods to find the Inverse: • Reflect graph over the line y = x. • Swapping x and y-values • Solving Algebraically: a. Set the function = yb.  Swap the x and y variablesc.  Solve for y

  7. Reflect the graph over y = x Graph original function f(x) = 2x + 3 It is drawn in blue. If reflected over the identity line, y = x, the original function becomes the red dotted graph. 

  8. Swapping x and y-values Given relation, find the inverse relation.

  9. Solving Algebraically Find the equation of the inverse of the relation f(x) = 2x – 4. 1. Set the function = y y = 2x – 4 • Swap the x and y variables x = 2y -4 • Solve for y y = (x + 4)/2

  10. Find the equation for the Inverse function. f(x) = -2x + 5 f-1(x) = -(x – 5)/2

  11. Verifying Inverse Functions Verify that f(x) = 2x – 4 and f-1(x) = ½x + 2 are inverses.

  12. Composition Functions (Inverse Functions)Part 2 Goal: Evaluate composition functions and prove functions are inverses of each other using composition functions.

  13. This "DO" and "UNDO" process can be stated as a composition of functions. • If functions f and g are inverse functions, f(g(x)) = g(f(x)) = x. • Example: If f(x) = x-1 and g(x) = x +1 then f(g(x)) = x and g(f(x)) = x • Think of them as "undoing" one another and leaving you right where you started.

  14. Verify that f and g are inverse functions. f(x) = 3x – 1 and g(x) = ⅓x + ⅓

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